Hardy-Littlewood circle method

The circle method represents a weighted additive count as an integral of products of exponential sums against a character. Near rationals with small denominator, major arcs produce a structured main term involving a singular series and an archimedean singular integral. Away from those rationals, minor arcs require cancellation bounds. The partition and approximation estimates depend on the problem; a formal integral alone is not a positivity proof.

Every sufficiently large odd integer is a sum of three primes, with repetitions allowed. The weighted count is asymptotic to $\mathfrak{S}(n)n^2/2$. Siegel–Walfisz theorem controls the major arcs, and Type I/II estimates for the prime exponential sum make the minor arcs negligible. Removing prime-power contributions gives genuine prime representations. This is the sufficiently-large theorem, not by itself a proof for every small odd integer.

A singular series assembles the local congruence densities in an additive counting asymptotic. For three prime summands it is ${\sum }_{q\ge 1}\mu (q)c_q(n)/\phi (q{)}^3$, with Ramanujan sum $c_q(n)$. Its Euler product vanishes for even $n$ and has a uniform positive lower bound for odd $n$. Different additive problems have different local factors, so this is a specific instance rather than a universal formula.

Minor arcs are the complement of the specified major arcs. They contribute an error only when a suitable pointwise or mean-value cancellation estimate is proved; their name does not imply that their measure is small.

In a circle-method problem of scale $n$, major arcs are neighborhoods of reduced rational points $a/q$ with small denominator, where the relevant exponential sum is approximated using distribution in residue classes. For the three-prime problem one may take $q\le (\log n{)}^B$ and $|\theta -a/q|\le (\log n{)}^B/n$, with suitably fixed $B$.